The table shows the concentration of a reactant in the reaction mixture over a period of time.

[tex]\[
\begin{tabular}{|l|l|}
\hline
\text{Time} & \text{Concentration} \\
\hline
$0 \, \text{s}$ & $1.8 \, \text{M}$ \\
\hline
$210 \, \text{s}$ & $1.2 \, \text{M}$ \\
\hline
$450 \, \text{s}$ & $0.8 \, \text{M}$ \\
\hline
$580 \, \text{s}$ & $0.6 \, \text{M}$ \\
\hline
$720 \, \text{s}$ & $0.4 \, \text{M}$ \\
\hline
\end{tabular}
\][/tex]

What is the average rate of the reaction over the first 450 seconds?

A. [tex]$2.2 \times 10^{-3} \, \text{M/s}$[/tex]
B. [tex]$1.9 \times 10^{-3} \, \text{M/s}$[/tex]
C. [tex]$2.0 \times 10^{-3} \, \text{M/s}$[/tex]
D. [tex]$1.7 \times 10^{-3} \, \text{M/s}$[/tex]



Answer :

To determine the average rate of the reaction over the first 450 seconds, follow these detailed steps:

1. Identify the initial concentration and the concentration at 450 seconds:
- Initial concentration at [tex]\( t = 0 \)[/tex] seconds is [tex]\( 1.8 \, M \)[/tex].
- Concentration at [tex]\( t = 450 \)[/tex] seconds is [tex]\( 0.8 \, M \)[/tex].

2. Calculate the change in concentration ([tex]\(\Delta[\text{Reactant}]\)[/tex]) over the time period:
[tex]\[ \Delta[\text{Reactant}] = [\text{Concentration at } 450 \text{ s}] - [\text{Concentration at } 0 \text{ s}] \][/tex]
[tex]\[ \Delta[\text{Reactant}] = 0.8 \, M - 1.8 \, M = -1.0 \, M \][/tex]

3. Determine the time interval ([tex]\(\Delta t\)[/tex]) over which this change occurs:
[tex]\[ \Delta t = 450 \, \text{seconds} - 0 \, \text{seconds} = 450 \, \text{seconds} \][/tex]

4. Calculate the average rate of the reaction using the formula:
[tex]\[ \text{Average rate} = \frac{\Delta[\text{Reactant}]}{\Delta t} \][/tex]
[tex]\[ \text{Average rate} = \frac{-1.0 \, M}{450 \, \text{seconds}} = -\frac{1.0}{450} \, M/\text{second} \approx -0.002222 \, M/\text{second} \][/tex]

Since we often give the rate of the reaction in positive terms (despite the fact that the concentration is decreasing), we can simply express the magnitude:

[tex]\[ \text{Average rate} = 0.002222 \, M/\text{second} \][/tex]

5. Compare our result with the given options:
[tex]\[ -0.002222 \approx -\frac{2.2}{1000} = -2.2 \times 10^{-3} \][/tex]

So, the closest option to our calculation is:
[tex]\[ 2.2 \times 10^{-3} \][/tex]

Thus, the average rate of the reaction over the first 450 seconds is [tex]\( 2.2 \times 10^{-3} \, M/\text{second} \)[/tex].